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''spin representation''

The simplest faithful linear representation (cf. [[Faithful representation|Faithful representation]]; [[Linear representation|Linear representation]]) of the [[Spinor group|spinor group]]  $   \mathop{\rm Spin} _ {n} ( Q) $, 
or the linear representation of the corresponding even Clifford algebra  $  C  ^ {+} = C  ^ {+} ( Q) $(
see [[Spinor group|Spinor group]];  $  Q $
is a [[Quadratic form|quadratic form]]). If the ground field  $  K $
is algebraically closed, then the algebra  $  C  ^ {+} $
is isomorphic to the complete matrix algebra  $  M _ {2  ^ {m}  } ( K) $(
where  $  n = 2 m + 1 $) 
or to the algebra  $  M _ {2  ^ {m-1} } ( K) \oplus M _ {2  ^ {m-1} } ( K) $(
where  $  n = 2 m $). 
Therefore there is defined a linear representation  $  \rho $
of the algebra  $  C  ^ {+} $
on the space of dimension  $  2  ^ {m} $
over  $  K $; 
this representation is called a spinor representation. The restriction of  $  \rho $
to  $   \mathop{\rm Spin} _ {n} ( Q) $
is called the spinor representation of  $   \mathop{\rm Spin} _ {n} ( Q) $. 
For odd  $  n $, 
the spinor representation is irreducible, and for even  $  n $
it splits into the direct sum of two non-equivalent irreducible representations  $  \rho  ^  \prime  $
and  $  \rho  ^ {\prime\prime} $, 
which are called half-spin (or) representations. The elements of the space of the spinor representation are called spinors, and those of the space of the half-spinor representation half-spinors. The spinor representation of the spinor group  $   \mathop{\rm Spin} _ {n} $
is self-dual for any  $  n \geq  3 $, 
whereas the half-spinor representations  $  \rho  ^  \prime  $
and  $  \rho  ^ {\prime\prime} $
of the spinor group  $   \mathop{\rm Spin} _ {2m} $
are self-dual for even  $  m $
and dual to one another for odd  $  m $. 
The spinor representation of  $   \mathop{\rm Spin} _ {n} $
is faithful for all  $  n \geq  3 $, 
while the half-spinor representations of  $   \mathop{\rm Spin} _ {2m} $
are faithful for odd  $  m $, 
but have a kernel of order two when  $  m $
is even.

For a quadratic form  $  Q $
on a space  $  V $
over some subfield  $  k \subset  K $, 
the spinor representation is not always defined over  $  k $. 
However, if the Witt index of  $  Q $
is maximal, that is, equal to  $  [ n / 2 ] $(
in particular, if  $  k $
is algebraically closed), then the spinor and half-spinor representations are defined over  $  k $. 
In this case these representations can be described in the following way if  $   \mathop{\rm char}  k \neq 2 $(
see [[#References|[1]]]). Let  $  L $
and  $  M $
be  $  k $-
subspaces of the  $  k $-
space  $  V $
that are maximal totally isotropic (with respect to the symmetric bilinear form on  $  V $
associated with  $  Q $) 
and let  $  L \cap M = 0 $. 
Let  $  C _ {L} $
be the subalgebra of the Clifford algebra  $  C = C ( Q) $
generated by the subspace  $  L \subset  V $, 
and let  $  e _ {M} \in C $
be the product of  $  m $
vectors forming a  $  k $-
basis of  $  M $. 
If  $  n $
is even,  $  n = 2m $, 
then the spinor representation is realized in the left ideal  $  C e _ {M} $
and acts there by left translation:  $  \rho ( s) x = s x $(
$  s \in C  ^ {+} $, 
$  x \in C e _ {M} $). 
Furthermore, the mapping  $  x \mapsto x e _ {M} $
defines an isomorphism of vector spaces  $  C _ {L} \rightarrow C e _ {M} $
that enables one to realize the spinor representation in  $  C _ {L} $, 
which is naturally isomorphic to the exterior algebra over  $  L $. 
The half-spinor representations  $  \rho  ^  \prime  $
and  $  \rho  ^ {\prime\prime} $
are realized in the  $  2  ^ {m-1} $-
dimensional subspaces  $  C _ {L} \cap C  ^ {+} $
and  $  C _ {L} \cap C  ^ {-} $.

If  $  n $
is odd, then  $  V $
can be imbedded in the  $  ( n + 1 ) $-
dimensional vector space  $  V _ {1} = V \oplus k \epsilon $
over  $  k $. 
One defines a quadratic form  $  Q _ {1} $
on  $  V _ {1} $
by putting  $  Q _ {1} ( v + \epsilon ) = Q ( v) - \lambda  ^ {2} $
for all  $  v \in V $
and  $  \lambda \in k $. 
$  Q _ {1} $
is a non-degenerate quadratic form of maximal Witt index defined over  $  k $
on the even-dimensional vector space  $  V _ {1} $. 
The spinor representation of  $  C  ^ {+} ( Q) $(
or of  $   \mathop{\rm Spin} _ {n} ( Q) $) 
is obtained by restricting any of the half-spinor representations of  $  C  ^ {+} ( Q _ {1} ) $(
or of  $   \mathop{\rm Spin} _ {n+} 1 ( Q _ {1} ) $) 
to the subalgebra  $  C  ^ {+} ( Q) $(
or  $   \mathop{\rm Spin} _ {n} ( Q) $, 
respectively).

The problem of classifying spinors has been solved when  $  3 \leq  n \leq  14 $
and  $  k $
is an algebraically closed field of characteristic 0 (see [[#References|[4]]], [[#References|[8]]], [[#References|[9]]]). The problem consists of the following: 1) describe the orbits of  $  \rho (  \mathop{\rm Spin} _ {n} ) $
in the spinor space by giving a representative of each orbit; 2) calculate the stabilizers in  $   \mathop{\rm Spin} _ {n} $
of each of these representatives; and 3) describe the algebra of invariants of the linear group  $  \rho (  \mathop{\rm Spin} _ {n} ) $.

The existence of spinor and half-spinor representations of the Lie algebra  $  \mathfrak s \mathfrak p _ {n} $
of  $   \mathop{\rm Spin} _ {n} $
was discovered by E. Cartan in 1913, when he classified the finite-dimensional representations of simple Lie algebras [[#References|[6]]]. In 1935, R. Brauer and H. Weyl described spinor and half-spinor representations in terms of Clifford algebras [[#References|[5]]]. P. Dirac [[#References|[3]]] showed how spinors could be used in quantum mechanics to describe the rotation of an electron.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Algèbre" , ''Eléments de mathématiques'' , Hermann (1970) pp. Chapt. II. Algèbre linéaire {{MR|0274237}} {{ZBL|0211.02401}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H. Weyl, "Classical groups, their invariants and representations" , Princeton Univ. Press (1946) (Translated from German) {{MR|0000255}} {{ZBL|1024.20502}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> P.A.M. Dirac, "Principles of quantum mechanics" , Clarendon Press (1958) {{MR|2303789}} {{MR|0116921}} {{MR|0023198}} {{MR|1522388}} {{ZBL|0080.22005}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> V.L. Popov, "Classification of spinors of dimension fourteen" ''Trans. Moscow Math. Soc.'' , '''1''' (1980) pp. 181–232 ''Trudy Moskov. Mat. Obshch.'' , '''37''' (1978) pp. 173–217 {{MR|0514331}} {{ZBL|0443.20038}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> R. Brauer, H. Weyl, "Spinors in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086770/s08677098.png" />-dimensions" ''Amer. J. Math.'' , '''57''' : 2 (1935) pp. 425–449 {{MR|1507084}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> E. Cartan, "Les groupes projectifs qui ne laissant invariante aucune multiplicité plane" ''Bull. Soc. Math. France'' , '''41''' (1913) pp. 53–96</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> C. Chevalley, "The algebraic theory of spinors" , Columbia Univ. Press (1954) {{MR|0060497}} {{ZBL|0057.25901}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> V. Gatti, E. Viniberghi, "Spinors in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086770/s08677099.png" />-dimensional space" ''Adv. Math.'' , '''30''' : 2 (1978) pp. 137–155</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> J.I. Igusa, "A classification of spinors up to dimension twelve" ''Amer. J. Math.'' , '''92''' : 4 (1970) pp. 997–1028 {{MR|0277558}} {{ZBL|0217.36203}} </TD></TR></table>